A vibration damping apparatus which is particularly applicable to an automotive engine mount and capable of fully absorbing vibration from vibration generating sources. A sleeve is secured to the automotive engine unit, and a housing is secured to the vehicle body. Once the vibration generated by the automotive engine is transmitted to the sleeve, liquid loaded in a main liquid chamber flows to a sub liquid chamber via a plurality of restricting passages. Vibration is effectively absorbed by the transit resistance against liquid flowing through those restricting passages. Independent of these restricting passages, a cylindrical member is provided which penetrates through the housing and a partition wall block. The cylindrical member is disposed in the path of the restricting passages providing connection between the main and sub liquid chambers. A hollow rotor is rotatably provided, inside of and on the same axial line as the cylindrical member, for opening and closing the restricting passages. When low-frequency vibration is generated, only the restricting passage devoid of the hollow rotor provides connection between the main and sub liquid chambers. Conversely, when high-frequency vibration is generated, those restricting passages each containing the hollow rotor and having a large sectional area are released to provide connection between both chambers thus reliably and effectively absorbing vibrations throughout an extensive range of frequencies.